To solve this problem using the Empirical Rule (also known as the 68-95-99.7 Rule), we need to understand how it applies to a normal distribution. The Empirical Rule states the following for a normal distribution:

1. 68% of the data falls within 1 standard deviation of the mean.
2. 95% of the data falls within 2 standard deviations of the mean.
3. 99.7% of the data falls within 3 standard deviations of the mean.

Given:

• Mean ($\mu$) = 150
• Standard deviation ($\sigma$) = 25

Let's break down the range of scores step by step:

Step 1: Define the relevant values

• We want to know the percentage of students who scored between 100 and 175.
• We will first express these values in terms of how many standard deviations they are from the mean.

1. 100 is how far from the mean: $100 = 150 - 2 \times 25 = 150 - 50$ So, 100 is 2 standard deviations below the mean.

2. 175 is how far from the mean: $175 = 150 + 1 \times 25 = 150 + 25$ So, 175 is 1 standard deviation above the mean.

Step 2: Apply the Empirical Rule

• From the Empirical Rule:
  • The range between 1 standard deviation below the mean and 1 standard deviation above the mean ($\mu - 1\sigma$ to $\mu + 1\sigma$) contains 68% of the data.
  • The range between 2 standard deviations below the mean and 2 standard deviations above the mean ($\mu - 2\sigma$ to $\mu + 2\sigma$) contains 95% of the data.

Now, we're interested in the range between 2 standard deviations below (100) and 1 standard deviation above (175). To calculate this:

1. From $\mu - 2\sigma$ to $\mu + 2\sigma$, we have 95% of the data.
2. From $\mu - 1\sigma$ to $\mu + 1\sigma$, we have 68% of the data.
3. The percentage of data between $\mu - 2\sigma$ (100) and $\mu - 1\sigma$ (125) can be calculated as: $\frac{95\% - 68\%}{2} = 13.5\%$

Thus, the percentage of students scoring between 100 and 175 is the sum of the following:

• The percentage between $\mu - 2\sigma$ (100) and $\mu - 1\sigma$ (125): 13.5%
• The percentage between $\mu - 1\sigma$ (125) and $\mu + 1\sigma$ (175): 68%

$13.5\% + 68\% = 81.5\%$

Final Answer:

Approximately 81.5% of students scored between 100 and 175.

Would you like further clarification or additional details?

Here are some related questions you might want to explore:

1. What is the probability that a student scores between 125 and 150?
2. What percentage of students scored above 175?
3. How many standard deviations above the mean is a score of 200?
4. What is the probability that a student scores between 150 and 200?
5. If the test scores are approximately normal, how would you find the probability of a student scoring below 100?

Tip: The Empirical Rule is a handy tool for quick estimates, but for exact probabilities, using the Z-score and standard normal distribution tables would give more precise results.